Download Density-Based Clustering Method

Density-Based Clustering Algorithms
Presented by: Iris Zhang
17 January 2003
Outline
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Clustering
Density-based clustering
DBSCAN
DENCLUE
Summary and future work
Clustering
Problem description
 Given:
A data set of N data items which are ddimensional data feature vectors.
 Task:
Determine a natural, useful partitioning of the
data set into a number of clusters (k) and noise.
Major Types of Clustering Algorithms
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Partitioning:
Partition the database into k clusters which are
represented by representative objects of them
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Hierarchical:
Decompose the database into several levels of
partitioning which are represented by dendrogram
Other kinds of Clustering Algorithms
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Density-based: based on connectivity and
density functions
Grid-based: based on a multiple-level granularity
structure
Model-based: A model is hypothesized for each
of the clusters and the idea is to find the best fit of
that model to each other
Density-Based Clustering
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A cluster is defined as a connected dense
component which can grow in any
direction that density leads.
Density, connectivity and boundary
Arbitrary shaped clusters and good
scalability
Two Major Types of Density-Based
Clustering Algorithms
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Connectivity based:
DBSCAN, GDBSCAN, OPTICS and
DBCLASD
Density function based:
DENCLUE
DBSCAN [Ester et al.1996]
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Clusters are defined as Density-Connected
Sets (wrt. Eps, MinPts)
Density and connectivity are measured by
local distribution of nearest neighbor
Target low dimensional spatial data
DBSCAN
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Definition 1: Eps-neighborhood of a point
NEps(p) = {q ∈D | dist(p,q) ≤ Eps}
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Definition 2: Core point
|NEps(q)| ≥ MinPts
DBSCAN
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Definition 3: Directly density-reachable
A point p is directly density-reachable from a
point q wrt. Eps, MinPts if
1) p ∈ NEps(q) and
2) |NEps(q)| ≥ MinPts (core point condition).
DBSCAN
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Definition 4: Density-reachable
A point p is density-reachable from a point q wrt. Eps and
MinPts if there is a chain of points p1, ..., pn, p1 = q, pn = p
such that pi+1 is directly density-reachable from pi
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Definition 5: Density-connected
A point p is density-connected to a point q wrt. Eps and
MinPts if there is a point o such that both, p and q are
density-reachable from o wrt. Eps and MinPts.
DBSCAN
DBSCAN
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Definition 6: Cluster
Let D be a database of points. A cluster C wrt. Eps and
MinPts is a non-empty subset of D satisfying the
following conditions:
1) ∀ p, q: if p ∈ C and q is density-reachable from p wrt.
Eps and MinPts, then q ∈ C. (Maximality)
2) ∀ p, q ∈ C: p is density-connected to q wrt. Eps and
MinPts. (Connectivity)
DBSCAN
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Definition 7: Noise
Let C1 ,. . ., Ck be the clusters of the database D wrt.
parameters Epsi and MinPtsi, i = 1, . . ., k. Then we define
the noise as the set of points in the database D not
belonging to any cluster Ci , i.e. noise = {p ∈D | ∀ i: p
Ci}.
DBSCAN
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Lemma 1:Let p be a point in D and |NEps(p)| ≥
MinPts. Then the set O = {o | o ∈D and o is
density-reachable from p wrt. Eps and MinPts} is
a cluster wrt. Eps and MinPts.
Lemma 2: Let C be a cluster wrt. Eps and MinPts
and let p be any point in C with |NEps(p)| ≥
MinPts. Then C equals to the set O = {o | o is
density-reachable from p wrt. Eps and MinPts}.
DBSCAN
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For each point, DBSCAN determines the
Eps-environment and checks whether it
contains more than MinPts data points
DBSCAN uses index structures (such as
R*-Tree) for determining the Epsenvironment
DBSCAN
Arbitrary shape clusters found by DBSCAN
DENCLUE [Hinneburg & Keim.1998]
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Clusters are defined according to the point
density function which is the sum of
influence functions of the data points.
It has good clustering in data sets with
large amounts of noise.
It can deal with high-dimensional data sets.
It is significantly faster than existing
algorithms
DENCLUE
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Influence Function:
Influence of a data point in its neighborhood
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Density Function:
Sum of the influences of all data points
DENCLUE
Definition 1:Influence Function
The influence of a data point y at a point x in
the data space is modeled by a function
f By : F d  R0
e.g.:
f Gauss ( x, y )  e
d ( x, y )2

2 2
DENCLUE
Definition 2:Density Function
The density at a point x in the data space is defined
as the sum of influences of all data points x
N
f BD ( x)   f Bxi ( x)
i 1
e.g.:
f
D
Gauss
N
( x)   e
i 1

d ( x , xi ) 2
2 2
DENCLUE
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Example
DENCLUE
Definition 3: Gradient
The gradient of a density function is defined as
N
f ( x)   ( xi  x)  f ( x)
D
B
e.g.:
f
i 1
N
D
Guass
xi
B
( x)   ( xi  x) e
i 1

d ( x , xi ) 2
2 2
DENCLUE
Definition 4: Density Attractor
A point x* ∈Fd is called a density attractor for a
given influence function, iff x* is a local maximum
of the density-function
Example of Density-Attractor
DENCLUE
Definition 5: Density attracted point
A point x* ∈Fd is density attracted to a density
attractor x*, iff  k ∈N: d(xk,x*)   with
-xi is a point in the path between x and its attractor x*
-density-attracted points are determined by a gradient-based
hill-climbing method
DENCLUE
Definition 6: Center-Defined Cluster
A center-defined cluster with density-attractor x*
( f BD (x*)   ) is the subset of the database which is
density-attracted by x*.
DENCLUE
Definition 7:Arbitrary-shaped cluster
A arbitrary-shaped cluster for the set of density-attractors X is
a subset C D,where
1) xC,x*  X: f BD (x*)   x is density attracted to x* and
2) x1*,x2*X:  a path P Fd from x1* to x2* with pP:
f BD ( p)  
DENCLUE
Noise-Invariance
Assumption:Noise is uniformly distributed in the data space
Lemma:The density-attractors do not change when the noise
level increases.
Idea of the Proof:
- partition density function into signal and noise
f ( x)  f
D
Dc
( x)  f ( x)
N
- density function of noise approximates a constant.
DENCLUE
Example of noise invariance
DENCLUE
Parameter-σ:
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It describes the influence of a data point in the data space.
It determines the number of clusters.
DENCLUE
Parameter-σ:
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Choose σ such that number of density attractors is constant
for the longest interval of σ.
DENCLUE
Parameter- ξ
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It describes whether a density-attractor is significant,
helping reduce the number of density-attractors such that
improving the performance.
DENCLUE
Experiment
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Polygonal CAD data (11-dimensional feature vectors)
Comparison between DBSCAN and DENCLUE
DENCLUE
DENCLUE
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Molecular biology to determine the behavior of the
molecular in the conformation space (19-dimensional
dihedral angle space with large amount of noise)
Folded State
Unfolded State
Folded Conformation of the Peptide
Summary
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arbitrary shaped clusters
good scalability
explicit definition of noise
noise invariance
high dimensional clustering
Future work
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Using density-based clustering method to
deal with high dimensional dataset
References
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[EKS+ 96] M. Ester, H-P. Kriegel, J. Sander, X. Xu, A Density-Based
Algorithm for Discovering Clusters in Large Spatial Databases with
Noise, Proc. 2nd Int. Conf. on Knowledge Discovery and Data
Mining, 1996.
[HK 98] A. Hinneburg, D.A. Keim, An Efficient Approach to
Clustering in Large Multimedia Databases with Noise, Proc. 4th Int.
Conf. on Knowledge Discovery and Data Mining, 1998.
[XEK+ 98] X. Xu, M. Ester, H-P. Kriegel and J. Sander., A
Distribution-Based Clustering Algorithm for Mining in Large Spatial
Databases, Proc. 14th Int. Conf. on Data Engineering (ICDE’98),
Orlando, FL, 1998, pp. 324-331.
References
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J. Sander, M. Ester, H-P. Kriegel, X. Xu, Density-Based Clustering in
Spatial Databases: the Algorithm GDBSCAN and its Applications,
Knowledge Discovery and Data Mining, an International Journal, Vol.
2, No. 2, Kluwer Academic Publishers, 1998, pp. 169-194.
Ankerst, M., Breunig, M., Kriegel, H.-P., and Sander, J. OPTICS:
Ordering Points To Identify . In Proceedings of ACM SIGMOD
International Conference on Management of Data, Philadelphia, PA,
1999.
Hinneburg A., Keim D. A.: Clustering Techniques for Large Data
Sets: From the Past to the Future ,Tutorial, Proc. Int. Conf. on
Principles and Practice in Knowledge Discovery (PKDD'00), Lyon,
France, 2000.
Q&A